ChE 503 A. Z. Panagiotopoulos 1


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1 ChE 503 A. Z. Panagiotopoulos 1 STATISTICAL MECHANICAL ENSEMLES 1 MICROSCOPIC AND MACROSCOPIC ARIALES The central question in Statistical Mechanics can be phrased as follows: If particles (atoms, molecules, electrons, nuclei, or even living cells) obey certain microscopic laws with specified interparticle interactions, what are the observable properties of a macroscopic system containing a large number of such particles? Examples of microscopic and macroscopic variables are given below for 1/6th mole of an n component monoatomic gas ( 10 3 molecules) obeying classical mechanics. Microscopic variables 3x10 3 positions (x,y,z) 3x10 3 velocities (x,y,z) Macroscopic variables n+ independent thermodynamic variables  e.g. for n1, N,,T PHASE SPACE The multidimensional space defined by the microscopic variables of a system. In the example above, it would be a 6x10 3 dimensional space, with independent variables (r N, p N ) (r 1,r,r 3,...,r N ;p 1,p,p 3,...,p N ) where bold symbols indicate vectors, r is the position and p the momentum (p mu). Momenta are used rather than velocities, because the classical and, especially, the quantum equations of change are more elegantly written in terms of momenta. The evolution of such a system in time is described by Newton's laws: d i dt r du ui ; i U ( r1, r, r3..., rn ) mi dt ri where U is the total potential energy of the system, the gradient of which is minus the force. For a much simpler system, a onedimensional harmonic oscillator, phase space is two dimensional, with coordinates the position and the momentum variables. ENSEMLES An ensemble is a collection of all microstates of a system, consistent with the constraints with which we characterize a system macroscopically. For example, a collection of all possible states of 1 Some material in this section is derived from Chap. 3, D. Chandler, "Introduction to modern statistical mechanics", Oxford University Press, 1987.
2 ChE 503 Statistical Mechanical Ensembles the 10 3 molecules of gas in the container of volume with a given total energy U is a statistical mechanical ensemble. For the onedimensional harmonic oscillator with a given energy, the phase space is a circular trajectory in (properly normalized) position and momentum space. ERGODIC HYPOTHESIS Experimental measurements on any macroscopic system are performed by observing a system for a finite period of time, during which the system samples a very large number of possible microstates. In order to connect the measured properties with the properties calculated from statistical mechanics, we have to assume that: For sufficiently long times, a macroscopic system will evolve through (or will come arbitrarily close to) all microscopic states consistent with the macroscopic constraints we impose in order to control the system. In other words, experimental measurements (performed by time averages) and ensemble averages are equivalent. The "ergodic hypothesis" is more than just a hypothesis. It is a general property of almost all real systems composed of a large number of particles. The ergodic hypothesis is equivalent to the postulate of classical thermodynamics that there exist stable equilibrium states fully characterized by n+ independent macroscopic variables. Note that the ergodic hypothesis does not state anything about the relative probabilities of observing given states  it just states that all states will eventually be observed. A general property F will thus obey the following rule: Fobserved Σ P F <F> [1] probability of finding the system in microstate value of property F in microstate ensemble average MICROCANONICAL ENSEMLE: CONSTANT U,,N asic Postulate of Statistical Mechanics: For an isolated system at constant U, and N, all microscopic states of a system are equally likely at thermodynamic equilibrium. If Ω(N,,U) is number of microstates with energy U, then the probability of microstate according to the postulate above, is v, P v 1/Ω (N,,U) [] The existence of a finite number of microstates for a given energy is a concept fully justified in quantum mechanics. In classical statistical mechanics, we can introduce a density of states by
3 ChE 503 Statistical Mechanical Ensembles 3 assuming that, say, two possible positions must be apart by more than a small distance in order to be counted as different. The (arbitrary) scales chosen to discretize positions and velocities determine how many states one counts in equation [], within a multiplicative constant. In other words, if one doubles the volume of a container, there are twice as many states for molecular positions. Example: alls in a box Let us consider a very simple example, namely a box with 10 slots, each containing a ball that can be at the bottom of the slot (with ener gy 0), or at one of two higher levels, with energies +1 and +, respectively. The number of microstates for this system depends on the overall energy of the box. There are: 10 1 state with energy 0; 10 states with energy +1; states with energy + and so on. As you can see, the number of microstates for even a simple system increases rapidly with the energy of the system. Definition of Entropy Let us define a quantity S, such that: S k lnω(n,,u) [3] where k is a constant (to be later identified as oltzmann's constant, k R/NA 1.380x103 J/K, where NA is Avogadro's Number, N A 6.03x10 3 mol 1 ). S has the following properties: 1. S is extensive: If we have two independent subsystems, A and, then S A+ k ln(ω A+ ) k ln(ω A Ω ) k lnω A + k lnω S A +S [4] The reason for this is that, for independent subsystems, each microstate of system A can be combined with a microstate of system to give a microstate of the combined system. For mixing two fluids, we only get the above expression if we assume that the particles in the systems are indistinguishable. If particles were distinguishable, additional "states" would be available to the combined system resulting from the possibility of exchanging the labels of particles. Although the indistinguishability of particles is really of quantum mechanical origin, it was introduced ad hoc by Gibbs before the development of quantum mechanics, in order to make statisticalmechanical entropy an extensive property.
4 ChE 503 Statistical Mechanical Ensembles 4. S is maximized at equilibrium: For a system with internal constraints (e.g. internal rigid walls or barriers to energy transfer), the number of possible microstates is always less than the number of microstates after the constraints are removed. S (N,,U) > S (N,,U; internal constraints) To see this second property, consider the box with particles of the example above, and think of any constraint to the system at a given energy (say U+). An example of a "constraint" would be to have that the first five slots have exactly 1 unit of energy. The number of microstates in this case is (5x55), less than the 55 states available to the unconstrained system. In conclusion, S has the same properties as the entropy. Statement () above is the microscopic statement of the Second Law of thermodynamics. From the Fundamental Equation of thermodynamics, du 1 P μi Td S Pd + μidni d S du + d dni i T T i T [5] The second form of the fundamental equation (the "Entropy Representation") is the most useful for Statistical Mechanics. Since (from equation 5) S 1 ln Ω 1 U N, T U N, k T β [6] The symbol β is commonly used in statistical mechanics to denote the inverse temperature. Example (cont.): alls in a box In the example above, we can now define the temperature of the box from equation 1. Initially, as the energy increases, the temperature increases, as expected. However, something strange happens to the system at energies greater than can you guess what that is? What is the sign of the temperature for energies > +10? Is this physically reasonable?
5 ChE 503 Statistical Mechanical Ensembles 5 CANONICAL ENSEMLE: CONSTANT N,,T Let us first obtain the equilibrium condition for two systems, I and, that are placed in thermal contact, so that they can exchange energy. The number of microstates available to the combined system must be a maximum at equilibrium, since the combined system is under conditions of constant N, and U discussed previously. Mathematically, the condition for equilibrium is that S total is maximum δs total 0 δs I + δs I S I S 0 δu + δu 0 I U U Since the total system is isolated, δu I + δu 0. Combining with the above condition, we obtain I S I U S U I T T [7] In other words, systems that can exchange energy much have the same temperature at equilibrium. Now, let us assume that one of the two systems is much larger than the other, so that it effectively acts as a "constanttemperature bath." The total system (small system + bath) is again considered under NU conditions. Now consider the small system at a given microstate with energy U. The energy of the bath is U U  U. The bath can be in any of Ω(U ) Ω( U  U ) microstates. Since the probability of the total system being in any particular combination of microstates with a given total energy is the same, the probability of finding the small system in state is P Ω (U  U ) exp (ln(ω (U  U )) [8] We can expand lnω around Ω(U) given that U is much smaller than U: ln Ω ln + U ( Ω( U U )) ln( Ω( U ) U higher order terms
6 ChE 503 Statistical Mechanical Ensembles 6 and substituting back in the expression for P using lnω/ U β 1/k T, P exp ( βu ) [9] This is a very important result. In words, we are finding that the probability of each microstate in the canonical ensemble (constant N,, T ) is proportional to the exponential of the energy divided by the temperature. In order to find the absolute probability of each microstate, we need to make sure that the sum of all the probabilities is one. The normalization constant for this is called the "canonical partition function," Q. Q( N,, T ) exp( β ) [10] all microstates i U i The summation over microstates is performed over all energies and particle positions. For the simple example of the box with particles we have been following, the partition function at a given inverse temperature β would be: Q(β) exp( β) + 55exp( β) + Where the first term in the summation comes from the single energy state with U 0, the second from the 10 states with U +1 and so on. For this simple system, both volume and number of particles are fixed, so they do not appear in the summation  however, in the general case the partition function would be a function of both N and. Once the partition function is defined, the probability of each microstate can now be written explicitly: P exp( βu ) Q [11] Therefore, in the canonical ensemble, a general property F is given by Fi exp( βu i ) all microstates i < F > exp( βu i ) [1] all microstates i It is possible to relate derivatives of lnq to thermodynamic properties. For example, ln Q ( Q / β) N, U exp( βu ) < U β N, Q exp( βu ) > [13] One can also calculate the averaged squared fluctuation of energy in the canonical ensemble:
7 ChE 503 Statistical Mechanical Ensembles 7 P U ( P ) ( δu ) ( U < U > < U > < U > U ( Q / β ) Q N, ( Q / β) Q N, ln Q β N, < U > β N, using the definition of the heat capacity, C < U >, we get T N, ( δ U ) k T [14] C This is a remarkable result! We have obtained a relationship between the thermodynamic parameters of a system and the size of the spontaneous fluctuations. It is interesting to note that the heat capacity of the system, C, is an extensive variable (grows linearly with the size of the system). This implies that the relative magnitude of the spontaneous fluctuations grows as: 1/ ( δu ) 1/ 1 < U > ( k T C ) < U > 1/ N N N [15] For a macroscopic system, N O(10 3 ), this is a very small number: the energy of an ideal gas system at equilibrium with a roomtemperature bath is constant to roughly 1 part in However, for small systems typically used in simulations, N 1001,000, so that typical fluctuations are 10%3%. Equation 15 is not valid near critical points, at which the heat capacity (and fluctuations) diverge. Q can be identified with a familiar thermodynamic function. To do this, let us write k ln Q k ln β exp( U ) [16] As just shown, the relative fluctuations in energy for a macroscopic system are very small. We can approximate the sum in equation 16 by summing just the dominant terms. All of these will have U <U>. There are Ω(<U>) such terms (since this is the number of microstates at that U, and thus k lnq k ln(ω (<U>) exp( β<u>)) k lnω (< U>) <U>/ T S U / T A / T [17] This should have been expected. In the microcanonical (const. NU) ensemble the important function Ω is such that k lnω S. Recall that S is the function maximized at const. NU. In the canonical (NT) ensemble, the thermodynamic function being minimized is A, or equivalently A / T is maximized. We see a similar relationship of A / T with k lnq.
8 ChE 503 Statistical Mechanical Ensembles 8 The approximation is exact at the "thermodynamic limit" (for an infinite system). The table below summarizes the connections between "classical" and "statistical" thermodynamic properties discussed thus far. Const. N, and U Const. N,, and T Classical Statistical Classical Statistical Entropy, S is maximized at equil. k lnω S Helmholtz Energy, A is minimized at equil. k lnq A / T GENERALIZED ENSEMLES AND LEGENDRE TRANSFORMS Starting from the fundamental equation in terms of S/k 1 k d S 1 P μ i du + d dn k T k T k T i [18] First Legendre Transform: y (1) S/k U/k T A/k T ( A is the Helmholtz free energy) y (0) S/k lnω y (1) A/k T lnq ariable Derivative ariable Derivative U N 1/(k T) β P/(k T) βp μ/(k T) βμ N 1/(k T) β U P/(k T) βp μ/(k T) βμ We then have all the relationships we need for the thermodynamic functions: 1 ln Ω β k T U, N [19] U lnq β, N [0] P k T lnq βp T, N ln Ω U, N [1]
9 ChE 503 Statistical Mechanical Ensembles 9 μ lnq βμ lnω k T N T, N U, [] One can continue this with Legendre transforms of higher order. The probabilities of microstates in the corresponding ensembles can be derived in a way completely analogous to the derivation for the canonical ensemble, involving a subsystem and bath of constant temperature. In general, starting from y(0) S/k ; y(n) y(0)  ξ1x 1  ξ X ξ n X n [3] where ξ i is the ith derivative of y(0) with respect to variable X i. The probability of a microstate in any given ensemble is proportional to P exp( ξ 1 X 1 ξ X... ξ n X n ) {where the ξ i s and X i s refer to y (0) } [4] with normalization factor Ξ ( ) exp ξ1 X 1 ξ X... ξn X n [5] Ξ is the partition function for the corresponding statistical ensemble. Using Ξ, the probability P v is now equal to: P v exp( ξ 1 X 1 ξ X... ξ n X n ) / Ξ [6] As for the canonical ensemble, Ξ has the property that : lnξ y (n) y (0) ξ 1 X 1 ξ X... ξ n X n [7] Using the expression above, it easy to prove the famous "Gibbs entropy formula": S k P ln P [8] Proof: P ln P P ( ln Ξ ξ X 1 ξ X... ξ 1 n X n ) {from equation 6} lnξ ξ 1 <X 1 > ξ <X > ξ n <X n > y (0) S / k {since all ξ i 's are constant in this ensemble}
10 ChE 503 Statistical Mechanical Ensembles 10 APPLICATION: GRAND CANONICAL ENSEMLE (CONST. μ,, T ) The probability of microstates in this ensemble is P exp( βu +βμn ) / Ξ [9] where lnξ S/k U/k T + μn/k T P/k T The derivative relationships for the Grand Canonical partition function Ξ can be obtained from a Legendre transform table: y (0) S/k lnω y () P/k T lnξ ariable Derivative ariable Derivative U N 1/(k T) β μ/(k T) βμ βμ P/(k T) βp 1/(k T) β U N P/(k T) βp e.g., ln Ξ ( βμ) < N > [30] Fluctuation formulae in the grand canonical ensemble are analyzed in the same fashion as in the canonical ensemble. For example, < ( δn) > < ( N < N > ) > < N > < N > P N P NP N ln Ξ < N > (βμ) [31] (βμ) β, β,
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